This work presents a hybrid analytical-numerical approach to evaluate the integral representations for the time-harmonic electromagnetic (EM) field components produced in the air space by a vertical magnetic dipole (VMD) placed on a plane homogeneous conducting medium. Explicit expressions for the fields are derived by substituting a rational approximation, generated by the vector fitting algorithm, for the non-analytic part of the integrand of the electric vector potential. This permits to rewrite the representation for the electric vector potential as a combination of simple closed-contour integrals around the pole singularities of the rational approximation, which may be directly evaluated. As a result, each field component is given as a sum of cylindrical Hankel functions depending on the radial distance between source and field points, plus an exponential term that is a function of the total distance of the field point from the dipole.
1. Zhdanov, M. S., Geophysical Electromagnetic Theory and Methods, Elsevier, Amsterdam, 2009.
2. Parise, M., "An exact series representation for the EM field from a circular loop antenna on a lossy half-space," IEEE Antennas and Wireless Propagation Letters, Vol. 13, 23-26, 2014. doi:10.1109/LAWP.2013.2296149
3. Farquharson, C. G., D. W. Oldenburg, and P. S. Routh, "Simultaneous 1D inversion of loop-loop electromagnetic data for magnetic susceptibility and electrical conductivity," Geophysics, Vol. 68, No. 6, 1857-1869, 2003. doi:10.1190/1.1635038
4. Parise, M., "Efficient computation of the surface fields of a horizontal magnetic dipole located at the air-ground interface," International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, Vol. 29, 653-664, 2016. doi:10.1002/jnm.2120
5. Wait, J. R., "Mutual electromagnetic coupling of loops over a homogeneous ground," Geophysics, Vol. 20, No. 3, 630-637, 1955. doi:10.1190/1.1438167
6. Beard, L. P. and J. E. Nyquist, "Simultaneous inversion of airborne electromagnetic data for resistivity and magnetic permeability," Geophysics, Vol. 63, No. 5, 1556-1564, 1998. doi:10.1190/1.1444452
7. Spies, B. R. and F. C. Frischknecht, "Electromagnetic sounding," Electromagnetic Methods in Applied Geophysics, Volume 2, 285-426, M. N. Nabighian, Ed., SEG, Tulsa, Oklahoma, 1988.
8. Parise, M., "Exact EM field excited by a short horizontal wire antenna lying on a conducting soil," AEU — International Journal of Electronics and Communications, Vol. 70, No. 5, 676-680, 2016. doi:10.1016/j.aeue.2016.02.004
9. Telford, W. M., L. P. Geldart, and R. E. Sheriff, Applied Geophysics, Cambridge University Press, New York, 1990. doi:10.1017/CBO9781139167932
10. Palacky, G. J., "Resistivity characteristics of geologic targets," Electromagnetic Methods in Applied Geophysics, Vol. 1, 52-129, Nabighian, M. N., Ed., SEG, Tulsa, Oklahoma, 1988.
11. Parise, M., "Full-wave analytical explicit expressions for the surface fields of an electrically large horizontal circular loop antenna placed on a layered ground," IET Microwaves, Antennas & Propagation, Vol. 11, 929-934, 2017. doi:10.1049/iet-map.2016.0590
12. Parise, M., "On the surface fields of a small circular loop antenna placed on plane stratified earth," International Journal of Antennas and Propagation, Vol. 2015, 1-8, 2015. doi:10.1155/2015/187806
13. Singh, N. P. and T. Mogi, "Electromagnetic response of a large circular loop source on a layered earth: A new computation method," Pure and Applied Geophysics, Vol. 162, 181-200, 2005. doi:10.1007/s00024-004-2586-2
14. Chew, W. C., Waves and Fields in Inhomogeneous Media, IEEE Press, Piscataway, NJ, 1995.
15. Parise, M. and G. Antonini, "On the inductive coupling between two parallel thin-wire circular loop antennas," IEEE Transactions on Electromagnetic Compatibility, Vol. 60, 1865-1872, 2018. doi:10.1109/TEMC.2018.2790265
16. Wait, J. R., "Fields of a horizontal loop antenna over a layered half-space," Journal of Electromagnetic Waves and Applications, Vol. 9, 1301-1311, 1995.
17. Parise, M., "A study on energetic efficiency of coil antennas used for RF diathermy," IEEE Antennas and Wireless Propagation Letters, Vol. 10, 385-388, 2011. doi:10.1109/LAWP.2011.2148190
18. Singh, N. P. and T. Mogi, "EMLCLLER-A program for computing the EM response of a large loop source over a layered earth model," Computer and Geosciences, Vol. 29, No. 10, 1301-1307, 2003. doi:10.1016/j.cageo.2003.08.008
19. Parise, M., V. Tamburrelli, and G. Antonini, "Mutual impedance of thin-wire circular loops in near-surface applications," IEEE Transactions on Electromagnetic Compatibility, Vol. 61, 558-563, 2019. doi:10.1109/TEMC.2018.2816030
20. Shastri, N. L. and H. P. Patra, "Multifrequency sounding results of laboratory simulated homogeneous and two-layer earth models," IEEE Trans. Geosci. Remote Sensing, Vol. 26, No. 6, 749-752, 1988. doi:10.1109/36.7706
21. Parise, M., "Fast computation of the forward solution in controlled-source electromagnetic sounding problems," Progress In Electromagnetics Research, Vol. 111, 119-139, 2011. doi:10.2528/PIER10101409
22. Kong, J. A., L. Tsang, and G. Simmons, "Geophysical subsurface probing with radio-frequency interferometry," IEEE Transactions on Antennas and Propagation, Vol. 22, No. 4, 616-620, 1974. doi:10.1109/TAP.1974.1140858
23. Singh, N. P. and T. Mogi, "Inversion of large loop transient electromagnetic data over layered earth models," Jour. Fac. Sci Hokkaido Univ. Ser. VII, Vol. 12, No. 1, 41-54, 2003.
24. Parise, M., "On the use of cloverleaf coils to induce therapeutic heating in tissues," Journal of Electromagnetic Waves and Applications, Vol. 25, 1667-1677, 2011. doi:10.1163/156939311797164945
25. Parise, M., "Improved Babylonian square root algorithm-based analytical expressions for the surface-to-surface solution to the Sommerfeld half-space problem," IEEE Transactions on Antennas and Propagation, Vol. 63, 5832, 2015. doi:10.1109/TAP.2015.2478958
26. Simons, N. R. S., A. Sebak, and G. E. Bridges, "Application of the TLM method to half-space and remote-sensing problems," IEEE Trans. Geosci. Remote Sensing, Vol. 33, No. 3, 759-767, 1995. doi:10.1109/36.387591
27. Singh, D., N. K. Choudhary, K. C. Tiwari, and R. Prasad, "Shape recognition of shallow buried metallic objects at x-band using ann and image analysis techniques," Progress In Electromagnetics Research B, Vol. 13, 257-273, 2009. doi:10.2528/PIERB09010301
28. Parise, M., "Quasi-static vertical magnetic field of a large horizontal circular loop located at the earth’s surface," Progress In Electromagnetics Research Letters, Vol. 62, 29-34, 2016. doi:10.2528/PIERL16053003
29. Ward, S. H. and G. W. Hohmann, "Electromagnetic theory for geophysical applications," Electromagnetic Methods in Applied Geophysics, Theory — Volume 1, 131-308, M. N. Nabighian (ed.), SEG, Tulsa, Oklahoma, 1988.
30. Kong, J. A., Electromagnetic Wave Theory, John Wiley & Sons, New York, 1986.
31. Parise, M., "Second-order formulation for the quasi-static field from a vertical electric dipole on a lossy half-space," Progress In Electromagnetics Research, Vol. 136, 509-521, 2013. doi:10.2528/PIER12112508
32. Parise, M., "A highly accurate analytical solution for the surface fields of a short vertical wire antenna lying on a multilayer ground," Waves in Random and Complex Media, Vol. 28, 49-59, 2018. doi:10.1080/17455030.2017.1319990
33. Parise, M., "An exact series representation for the EM field from a vertical electric dipole on an imperfectly conducting half-space," Journal of Electromagnetic Waves and Applications, Vol. 28, 932-942, 2014. doi:10.1080/09205071.2014.897653
34. Gustavsen, B. and A. Semlyen, "Rational approximation of frequency domain responses by vector fitting," IEEE Trans. Power Delivery, Vol. 14, No. 3, 1052-1061, 1999. doi:10.1109/61.772353